Math Insight

Two dimensional autonomous differential equation problems

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  1. For the following system of differential equations, plot the nullclines in the phase plane: \begin{align*} \diff{u}{t} &= -u(v-2)\\ \diff{v}{t} &= v(3-u). \end{align*} Find the equilibria and indicate their location on the phase plane. Plot direction arrows and a plausible solution in the phase plane for the initial conditions $r(0)=1$ and $s(0)=1$.

  2. Consider the two-dimensional system of autonomous differential equations \begin{align*} \diff{z}{t} &= z+v - 1\\ \diff{v}{t} &= z-v +1 \end{align*}
    1. Find the nullclines and sketch them on the phase plane. Be sure to label your axes and the nullclines.
    2. Identify all equilbria. Give their values and show them in the phase plane.
    3. Sketch a direction arrow in each of the regions of the phase plane divided by the nullclines.
    4. Sketch a direction arrow on each segment of each nullcline.
    5. Sketch a plausible solution in the phase plane starting at the initial condition $(z(0),v(0))=(0,0)$ that is consistent with your direction arrows.
    6. Sketch a plausible solution of $z(t)$ and $v(t)$ versus time $t$ that corresponds to your sketch in the phase plane.
    7. Estimate the solution $z(t)$ and $v(t)$ with initial conditions $(z(0),v(0))=(0,0)$ using three steps of the Forward Euler algorithm with time step $\Delta t = 1$.

  3. If $v(t)$ represents the voltage of a neuron at time time and $w(t)$ represents the state of its potassium channels at time $t$, then a model of their evolution is \begin{align*} \diff{v}{t} &= -v(v-a)(v-1) -w\\ \diff{w}{t} &= \varepsilon (v-\gamma w) \end{align*} where $v_0$ and $w_0$ are the initial conditions, $a$, $\varepsilon$, and $\gamma$ are parameters. We'll set $a=0.2$, $\varepsilon=0.006$, and $\gamma=1.6$.
    1. Calculate the nullclines and plot them on the phase plane. To help you, we show a phase plane plot below. Dotted lines are included as a hint, but they are not intended to be complete. Be sure to label the nullclines.
    2. Identify all equilibria. Give their values and show them in the phase plane.
    3. Sketch a direction arrow in each of the regions of the phase plane divided by the nullclines.
    4. Sketch a direction arrow on each segment of each nullcline.
    5. Suppose the neuron has received a small input so its voltage starts at $v(0)=0.16$ with potassium channels at $w(0)=0$. Sketch a plausible solution in the phase plane starting at that initial condition. The solution should be consistent with your direction arrows.
    6. Suppose the neuron has received a large input so its voltage starts at $v(0)=0.4$ with potassium channels at $w(0)=0$. Sketch a plausible solution in the phase plane starting at that initial condition. The solution should be consistent with your direction arrows.

  4. Consider the two-dimensional system of autonomous differential equations \begin{align*} \diff{v}{t} &= f(v,r)\\ \diff{r}{t} &= g(v,r), \end{align*} where $f$ and $g$ are some functions. The nullclines of this system are plotted in the below phase plane, where $v$ is plotted on the horizontal axis and $r$ is plotted on the vertical axis. The $v$-nullcline is plotted by the thin green line; the $g$-nullcline is plotted by the thick blue line. The nullclines divide the phase plane into four regions. In each of those regions, directions arrows are shown indicating the general direction of the flow of solution trajectories in the region. The direction arrows indicate whether the state variables ($v$ and $r$) increase or decrease in the region. Example phase plane with nullclines 1
    1. Along each segment of each nullcline, draw a direction arrow indicating the direction of solution trajectories that cross the nullcline.
    2. Identify all equilbria by indicating their position in the phase plane and estimating the values of $v$ and $r$ at each equilibrium.
    3. Imagine that the initial condition was $(v(0),r(0))=(2,-3)$, as shown by the red diamond. Sketch in the phase plane a plausible solution of the differential equations with that initial condition. You should should be consistent with all the information given and what you determined.
    4. Sketch how this solution might look like when plotted versus time. Plot a curve for both $v(t)$ and $r(t)$ where you use $t$ for the horizontal axis. You solution simply needs to be consistent with the solution you sketched in the phase plane.

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Elementary dynamical systems